Idea

Given an actionρ\rho of a groupGG on a setSS, the action groupoid S//GS//G is a bit like the quotient set S/GS/G (the set of GG-orbits). But, instead of taking elements of SS in the same GG-orbit as being equal in S/GS/G, in the action groupoid they are just isomorphic. We may think of the action groupoid as a resolution of the usual quotient. When the action of GG on SS fails to be free, the action groupoid is generally better-behaved than the quotient set.

The action groupoid also goes by other names, including ‘weak quotient’. It is a special case of a ‘pseudo colimit’, as explained below. It is also called a “semidirect product” and then written S⋊GS \rtimes G. The advantage of this is that it accords with the generalisation to the action of a group GG on a groupoid SS, which is relevant to orbit space considerations, since if GG acts on a space XX it also acts on the fundamental groupoid of XX; this is fully developed in “Topology and Groupoids”, Chapter 11.

Definition

In category theory

Given an actionρ:S×G→S\rho : S \times G \to S of a group GG on the set SS, the action groupoidS//GS//G (or, more precisely, S//ρGS//_\rho G) is the groupoid for which:

an object is an element of SS

a morphism from s∈Ss \in S to s′∈Ss' \in S is a group element g∈Gg \in G with gs=s′g s = s'. So, a general morphism is a pair (g,s):s→gs(g,s) : s \to g s.

Notice also that an action of GG on the set SS gives rise to a morphism p:S⋊G→Gp: S \rtimes G \to G which has the property of unique path lifting, or in other words is a discrete opfibration. It is also called a covering morphism of groupoids, and models nicely covering maps of spaces.

Higgins used this idea to lift presentations of a group GG to presentations of the covering morphism of GG derived from the action of GG on cosets, and so to apply graph theory to obtain old and new subgroup theorems in group theory.

As a stack

In the case where the action is internal to sets with structure, such as internal to Diff one wants to realize the action groupoid as a Lie groupoid. That Lie groupoid in turn may be taken to present a differentiable stack which then usually goes by the same name S//GS//G.

Properties

Relation to representation theory

The action groupoids X//GX//G of a group GG come equipped with a canonical map to BG≃*//G\mathbf{B}G \simeq \ast //G. Regarded via this map as objects in the slice of groupoids over BG\mathbf{B}G, action groupoids are in fact equivalent to the actions that they arise from.